It is not that difficult to compute the branches of .
The branches of are simply the fixed points of . In this way you can compute the branches via , where .
This is however valid only for as the numbering of the branches in this interval may be a bit different. For the branch 0 and -1 (which are non-real) join and for slightly smaller it splits again into two real branches.
The negative branches are simply the conjugates of the positive branches:.

Edit:
As it is so easy to compute it is perhaps more appropriate to compute from by instead of computing from .